Goldman-Hodgkin-Katz equation, reverse electrodialysis, and everything in between

In the past 80 years, the Goldman-Hodgkin-Katz (GHK) equation has been the gold-standard framework for interpreting countless biological and physiological experiments and simulations that involve ion transport in nanopores, nanochannels, and ion-channels subjected to a combined ionic concentration and electric potential gradients. In this work, we revisit the mathematical derivation used to develop the GHK model and show that this model is internally inconsistent when the Debye length is substantially smaller than the longitudinal length of the channel. In particular, we show that its infamous assumption of a constant electric field is incorrect, which leads to substantial errors, including the inability of this model to satisfy local and global electroneutrality. Then, leveraging key insights from the field of reverse electrodialysis (RED), we derive a new internally consistent model that does not assume that the electric field is constant and satisfies electroneutrality. This new model has several advantages. First, while the mathematics are substantially more complicated, the derivation does not include ad hoc assumptions, and the model is internally consistent. Second, the new solution connects the two realms of GHK and RED, which consider the same equations but in opposing limits, negligible or substantial surface charge density effects, respectively. Third, while the expressions for the new model are complicated, the new model can be reduced to several limits, which allows for a much easier and more straightforward analysis. Finally, all of our newly derived results show remarkable correspondence to nonapproximated numerical simulations. This work provides a brand-new framework for interpreting (and reinterpreting) ion transport experiments in any charge-selective system.